Question: Given $ m \angle RPS = 8x - 103$, $ m \angle QPR = 6x + 28$, and $ m \angle QPS = 135$, find $m\angle RPS$. $P$ $Q$ $S$ $R$
Answer: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Substitute in the expressions that were given for each measure: $ {6x + 28} + {8x - 103} = {135}$ Combine like terms: $ 14x - 75 = 135$ Add $75$ to both sides: $ 14x = 210$ Divide both sides by $14$ to find $x$ $ x = 15$ Substitute $15$ for $x$ in the expression that was given for $m\angle RPS$ $ m\angle RPS = 8({15}) - 103$ Simplify: $ {m\angle RPS = 120 - 103}$ So ${m\angle RPS = 17}$.